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Unformatted text preview: IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2004, Professor Whitt First Midterm Exam: Thursday, February 19 Chapters 1-4 in Ross, SOLUTIONS Justify your answers; show your work. 1. Satisfaction Survey ( 25 points) In its never-ending quest for educational excellence, the IEOR Department has tried a dif- ferent probability-and-statistics textbook for its introductory probability-and-statistics course during each of the last three semesters. During the first semester, 50 students used the textbook by Professor Mean; during the second semester, 30 students used the textbook by Professor Median; and during the third semester, 20 students used the textbook by Professor Mode. Surveys were taken at the end of each course, asking all students their opinion. In the first survey, 20 of the 50 students admitted being satisfied with Means book; in the second survey, 15 of the 30 students admitted being satisfied with Medians book; and in the third survey, 16 of the 20 students admitted being satisfied with Modes book. (a) What is the probability that a student selected at random from all these students will have admitted being satisfied by his textbook? It is helpful to start by constructing a probability tree . Let S denote satisfied; let N denote not satisfied. probability tree for textbooks Mean Mode Median S N S N S N random student 0.5 0.2 0.3 0.4 0.6 0.5 0.5 0.8 0.2 0.20 0.15 0.16 P ( S ) = P ( S Mean ) + P ( S Median ) + P ( S Mode ) = 0 . 20 + 0 . 15 + 0 . 16 = 0 . 51 (b) Suppose that a student, selected at random from all these students, admitted having been satisfied by his textbook. What is the probability that the student used the textbook by Professor Mean? Use Bayes Theorem , exploiting the definition of conditional probability : P ( Mean | S ) = P ( Mean S ) P ( S ) = . 20 . 20 + 0 . 15 + 0 . 16 = . 20 . 51 . 39 The last numerical calculation is not required. (c) By this experiment, which book is most likely to be best (assuming that we can judge by student opinion)? It is natural to pick the book with P ( S | book ) being largest. These probabilities are obtained from the second set of branches on the tree. By that criterion, the textbook by Professor Mode is most likely to be best: P ( S | Mode ) = 0 . 80, larger than the other two P ( S | book )....
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